Approximating Minimum Power Degree and Connectivity Problems

Zeev NutovThe Open University of Israel

Joint Work with: Guy Kortsarz

Vahab Mirrokni

Elena Tsanko

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Talk Outline

• Min-Power Problems - Motivation • Defining the Problems • Relations Between the Problems• Our Results• O(log n)-Approximation Algorithm for

Min-Power Edge-Multi-Cover (MPEMC)• 3/2-Approximation Algorithm for

Min-Power Edge-Cover (MPEC)

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The Cost Measure-Wired Networks:connecting every two nodes incurs a cost.

• Nodes in the network correspond to transmitters.

• More power larger transmission range.

• Transmission range = usually (but not always)

disk centered at the node.

The Power Measure-Wireless Networks:every node connects to all nodes in its “range”.

The Power Measure-Motivation

Goal: Find min-power range assignment so that

the resulting communication network satisfies

some prescribed property.

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b

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Range assignment Communication network

Example

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Relating powers and costs:

Directed: c(H)/Δ(H) ≤ p(H) ≤ c(H) (Δ(H)=max-outdegree)

Undirected: c(H)/√|F|/2 ≤ p(H) ≤ 2c(H)

c(H) ≤ p(H) ≤ 2c(H) if H is a forest

c(H) = n-1

p(H) = n

c(H) = n-1

p(H) = 1

directed undirected

Power vs Cost

Definition: Let H=(V,F) be a graph with edge-costs {c(e):eF}power of v in H: pF(v) = max{c(e):eF(v)} = maximum cost of an edge leaving vThe power of H: p(H) = pF(V)= ∑vV pF(v)

−−−

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Minimum Power Edge-Multi-Cover (MPEMC)Instance: A graph G = (V,E) with edge costs {c(e):e E}, and degree requirements {r(v):v V}.

Objective: Find a minimum power subgraph H of G so that H is an r-edge-cover.

Defining the ProblemsDefinition: Given a degree requirement function r on V, an edge set F on V is an r-edge-cover if degF(v) ≥ r(v) for all v V

Minimum Power k-Connected Subgraph (MPkCS)Instance: A graph G = (V,E) with edge costs {c(e):e E}, and an integer k. Objective: Find a minimum power k-connected spanning

subgraph H of G.

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k-1k-clique

(rmax+1)- Approximation for

MPEMC Algorithm: For every vV pick a set F(v) of r(v) cheapest edges incident to v.

Tight examplecosts = 1requirements: r(v)=k-1 for clique nodes.

opt = k (the clique edges)Algorithm : k·k (edges of the stars)

max max1 1 1 opt .Fv V v V

p V r v v r v r

Claim: The approximation ratio is (rmax+1) and this is tight.

Proof: Let π(v) = max{c(e):e F(v)}. Clearly, ΣvV π(v) ≤ opt.

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Relating Approximation Ratios

= approximation ratio for MPkCS = approximation for MPEMC with r(v)=k-1 for all vV ρ = approximation ratio for MCkCS

Currently, ρ = O(log k log n/(n-k))=O(log2k) [FL08,N08]

Corollary: =Θ() provided =O(ρ).

Theorem: ≤ 2 + [HKMN05, JKMWY05] ≤ 2 +1 [HKMN05] ≤ [LN07]

Previous best value of (and of ): O(log4n) [HKMN05]

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Our Result

Theorem 1MPEMC admits an O(log n)-approximation algorithm.Thus MPkCS admits an approximation algorithm with ratio O(log n + log k log n/(n-k)) = O(log n log n/(n-k)).

Previous ratio for MPEMC, MPkCS: O(log4n) [HKMN05].

What about MPEC, when we have 0,1 requirements?

Previous ratio for MPEC: 2.

Theorem 2MPEC admits a 3/2-approximation algorithm.

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Proof of Theorem 1

• Reduction to bipartite graphs • Algorithm: iteratively covers a constant fraction of the

total requirement with edge set of power ~ opt• Ignoring dangerous edges: Reduction to a special case of

Budgeted Multi-Coverage with Group Constraints problem

Remark: Standard greedy methods do not work, because:Claim: The “budgeted” version of MPEMC is harder than the Densest k-Subgraph problem.Proof: Given an instance G,k of DkS set: {c(e)=1: eE}, {r(v)=k-1: v V}, and budget P=k. In the budgeted MPEMC we seek a k-subgraph with maximum number of edges; this is exactly DkS.

Proof Outline

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Reduction to bipartite graphs

- approximation algorithm for bipartite MPEMC implies2 -approximation algorithm for general MPEMC.

auav

bvbu

A

B

u v

The Reduction: Given an instance G=(V,E),c,r of MPEMC,

construct an instance G'=(A+B,E′),c ',r ' of bipartite MPEMC:- each of A,B is a copy of V;

- for every uv E there are edges auav , avau with cost c(uv) each;

- r '(bv)=r(v) for bv B and r '(av)=0 for av in A.

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Algorithm for bipartite graphs

The Main Lemma: There exists a polynomial algorithm A that given an integer τ andγ > 1 either establishes that τ ≤ opt, or returns an edge set I so that:

(1) pI(V) ≤ (1+γ) τ (2) rI(B) ≤ (1-) r(B) =(1-1/e)(1-1/γ)

Definition: For an edge set I, the residual requirement of bB is: rI(b)=max{r(b)-degI(b),0}; let rI(B)=ΣbB rI(b).

The Algorithm: Initialization: F ← , γ ← 1/2While r(B) > 0 do:

• Find the smallest τ so that A returns I E satisfying (1),(2).• F ← F +I, E ← E–I, r←rI .

EndWhile

The approximation ratio: O(log r(B))=O(log n2)=O(log n).

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Proof of The Main Lemma

opt ( ) ( ( ) / ) ( )Jb D b D b D

p b r b R r b r DR R

Lemma 1: If τ ≥ opt then rJ(B) ≥ R(1-1/γ) for any set J of dangerous edges with pJ(B) ≤ τ. Thus:

- The dangerous edges in OPT cover at most R/γ of the demand; - The non-dangerous edges cover at least (1-1/γ)R of the demand.

Definition: Let R=r(B). An edge abE is dangerous if

c(ab) ≥ γτ · r(b)/R.

Proof: Let D={bB :degJ(b) ≥ 1}. Then

Lemma 2: pF(B) ≤ γτ for any set F of non-dangerous edges.

( ) ( ( ) / ) ( )F F

b B b B b D

r Dp B p b r b R r b

R R

Proof:

Thus r(D) ≤ R/γ, which implies rJ(B) ≥ R-r(D) ≥ R(1-1/γ)

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Finishing the Proof of The Main LemmaCorollary: If τ ≥ opt then the non-dangerous edges: - cover at least (1-1/γ)R of the demand; - incur power at most γτ at B.

Thus after the dangerous edges are ignored, we obtain the problem:

Problem (*) admits a (1-1/e)-approximation algorithm:The proof is slightly more complicated than the proof of [KMN99] that Budgeted Max-Coverage admits a (1-1/e)-approximation algorithm.

Algorithm A:1. Delete all dangerous edges.2. Let I be the edge set returned by the (1-1/e)-

approximation algorithm for Problem (*).3. If rI(B) ≤ (1-)R then return I; Else declare “τ ≤ opt”.

(*) max{r(B)-rI(B) : I E, pI(A) ≤ τ}

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A 3/2-Approximation Algorithm for MPEC

Minimum Power Edge-Cover (MPEC)

The idea behind the algorithm: Reduction to Min-Cost Edge-Cover (solvable in polynomial time) with loss of 3/2 in the approximation ratio.

1. For every u,v S compute a minimum {u,v}-cover I(uv) that consists of the edge uv or of two adjacent edges su,sv.

2. Construct an instance G′=(S,E′),c′ of Min-Cost Edge-Cover: G′ is a complete graph on S and c′(uv)=p(I(uv)).

3. Find a minimum-cost edge-cover I′ in G′,c′.

4. Return I = {I(uv) : uv I′}.

Instance: A graph G=(V,E), edge-costs {c(e):e E}, and S V.Objective: Find a minimum power S-cover I E.

U

Algorithm:

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Proof-Sketch: • Any inclusion minimal S-cover is a collection of stars.• Thus it is enough to consider the case when OPT is a star.• Recall Step 1 in the algorithm:

For every u,v S compute a minimum {u,v}-cover I(uv) that consists of one edge or of two adjacent edges.

• We prove: any star I with costs can be decompose into 2-stars and single edges (with at least one edge) so that: The sum of the powers of 2-stars and edges ≤ 3/2·p(I)

(i) If I′ is an edge cover in G′ then I covers S in G and p(I) ≤ c′(I′).

(ii) opt′ ≤ 3/2 · opt (opt′ = minimum-cost of an edge-cover in G′,c′)

The Main Lemma:

Approximation Ratio

The ratio 3/2 follows since: p(I) ≤ c′(I′) = opt′ ≤ 3/2 ·opt .

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For unit costs, p(I)=+1 and p() ≤ 3·/2+1, so p()/p(I) ≤ 3/2.

2-Decompositions of Stars

A 2-decomposition of a star I is a partition of I into 2-stars and edges (with at least one edge) that covers the nodes of I.

The power of is the sum of the powers of is parts.

Definition:

p(I) = 6p()=8

p(I) = 7p()=10

Lemma:

For general costs, any star I admits a 2-decomposition so that:

p() ≤ 3/2 · p(I)

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Summary and Open Questions

1. O(log n)-approximation for MPEMC.2. O(log n log n/(n-k))-approximation for MPkCS.3. 3/2-approximation for MPEC.

1. Constant ratio for MPEMC?2. (log n)-hardness for MPEMC?3. Approximation hardness of MPkCS/MCkCS…4. 4/3-approximation for MPEC?

Results:

Open Questions: